Introduction to sets

Unit 1 Introduction to sets

Key unit competence: Use set notations, Venn diagrams and relations in solving related problems.

Key ideas

• A set is a well-defined collection of distinct objects, named with a capital letter. The objects are its elements or members.
• $x\in A$ means "$x$ is an element of $A$"; $x\notin A$ means it is not. The cardinal number $n(A)$ is the number of elements of $A$.
• Special sets: the empty set $\varnothing$ or $\{\}$ has no element; a singleton has exactly one element; disjoint sets have no common element; a set is finite or infinite.
• $B\subset A$ ($B$ is a subset of $A$) when every element of $B$ is in $A$. The empty set is a subset of every set. A set with $n$ elements has $2^n$ subsets.
• Intersection $A\cap B$: elements in both sets. Union $A\cup B$: elements in at least one set (common elements written once). Difference $A-B$: elements of $A$ not in $B$. Symmetric difference $A\Delta B=(A-B)\cup (B-A)$.
• The universal set $U$ (or $\epsilon$) contains all elements under consideration. The complement $A^{\prime }$ contains the elements of $U$ not in $A$.
• Equal sets have exactly the same elements; equivalent sets have the same number of elements.
• A relation links elements of sets; it is a set of ordered pairs $(x,y)$, where order matters. The Cartesian product $A\times B=\{(x,y):x\in A,y\in B\}$.
• A relation on a set can be reflexive ($xRx$), symmetric ($xRy\Rightarrow yRx$) and transitive ($xRy$ and $yRz\Rightarrow xRz$). If it has all three it is an equivalence relation. A papygram shows a relation inside one set with arrows.
• A mapping associates each element of the domain with exactly one element of the range. A mapping is a function; it can be one-to-one or many-to-one.

Formulas to know

Common misconceptions

• "$\{0\}$ is the empty set." No: $\{0\}$ is a singleton containing the element 0; the empty set $\varnothing$ has no element at all.
• "$(2,5)$ and $(5,2)$ are the same pair." No: in an ordered pair the order matters, so $(2,5)\ne (5,2)$.
• "Equivalent sets are equal sets." No: equivalent sets only have the same number of elements; equal sets have the same elements.

Worked examples

Practice questions

Easier and harder questions are mixed, just like in a real exam.

1. List the elements of the set $G$ of vowels in the word "algebra" and give $n(G)$.

   Show answer

   $G=\{a,e\}$ (each element is written once), so $n(G)=2$.

   Common mistake: Writing $a$ twice: elements of a set are distinct.

2. A mapping is defined by $f(x)=4x$ on the domain $\{1,2,3\}$. Give the range and say whether the mapping is one-to-one.

   Show answer

   Range $=\{4,8,12\}$. Different inputs give different outputs, so it is one-to-one.

   Common mistake: Confusing domain (inputs) with range (outputs).

3. Given $M=\{1,3,5,7,9\}$ and $N=\{2,5,6,9\}$, find $M\cap N$, $M\cup N$ and $M-N$.

   Show answer

   $M\cap N=\{5,9\}$, $M\cup N=\{1,2,3,5,6,7,9\}$, $M-N=\{1,3,7\}$.

   Common mistake: Repeating 5 and 9 in the union.

4. Write the relation "is two less than" from $A=\{0,1,2,3\}$ to $B=\{2,3,4,5\}$ as a set of ordered pairs.

   Show answer

   $\{(0,2),(1,3),(2,4),(3,5)\}$.

   Common mistake: Reversing the pairs, e.g. writing $(2,0)$.

5. List all subsets of $\{x,y,z\}$.

   Show answer

   $\varnothing ,\{x\},\{y\},\{z\},\{x,y\},\{x,z\},\{y,z\},\{x,y,z\}$: there are $2^3=8$.

   Common mistake: Forgetting the empty set or the set itself.

6. Show that the relation "lives in the same district as" is an equivalence relation.

   Show answer

   Reflexive: everyone lives in the same district as himself. Symmetric: if $x$ lives in the same district as $y$, then $y$ lives in the same district as $x$. Transitive: if $x\sim y$ and $y\sim z$, all three share the district, so $x\sim z$.

   Common mistake: Checking only one or two of the three properties.

7. In a group of 50 traders, 32 sell beans, 25 sell maize and 4 sell neither. Draw a Venn diagram and find how many sell both.

   Show answer

   $50-4=46$ sell at least one. $32+25-x=46\Rightarrow x=11$. Beans only 21, maize only 14, both 11, neither 4.

   Common mistake: Forgetting to remove the 4 who sell neither before using the formula.

8. Are $R$ = letters of "Mississippi" and $Q$ = letters of "miss" equal sets?

   Show answer

   $R=\{m,i,s,p\}$ and $Q=\{m,i,s\}$. $p\in R$ but $p\notin Q$, so they are not equal (and not equivalent).

   Common mistake: Counting repeated letters as different elements.

Matching and drag-and-drop

1. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. In a papygram of the relation "is a multiple of" on $\{2,3,4,6\}$, which arrow is drawn?	a. From 6 to 3
   2. How many subsets does a set with 3 elements have?	b. Many-to-one
   3. "People → birth month" is an example of which kind of mapping?	c. 8
   4. $U=\{1,2,3,\dots ,10\}$ and $A=\{2,4,6,8,10\}$. What is $A^{\prime }$?	d. $\{1,3,5,7,9\}$

   Show answer

   1 → a (From 6 to 3) · 2 → c (8) · 3 → b (Many-to-one) · 4 → d ($\{1,3,5,7,9\}$)

   • 6 is a multiple of 3, so the arrow goes from 6 to 3. 3 is not a multiple of 6.
   • A set with $n$ elements has $2^n$ subsets: $2^3=8$, including $\varnothing$ and the set itself.
   • Each person has one birth month, but many people can share a month, so it is many-to-one.
   • The complement contains the elements of $U$ that are not in $A$: the odd numbers.

2. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   60  ·  16  ·  13  ·  reflexive and transitive but not symmetric
   1. If $n(A)=10$, $n(B)=6$ and $A$ and $B$ are disjoint, then $n(A\cup B)$ equals .
   2. The relation "is a factor of" on $\{2,4,8\}$ is .
   3. In a survey, $n(M)=22$, $n(S)=17$ and $n(M\cup S)=30$. How many belong to $M$ only? →
   Show answer

   1: 16 · 2: reflexive and transitive but not symmetric · 3: 13

   • Step 1. Disjoint sets have $n(A\cap B)=0$.
     Step 2. So $n(A\cup B)=10+6=16$.
   • Every number is a factor of itself (reflexive), $2|4$ and $4|8$ give $2|8$ (transitive), but $4$ is not a factor of 2 (not symmetric).
   • Step 1. $n(M\cap S)=22+17-30=9$.
     Step 2. So $M$ only $=22-9=13$.

3. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. $n(A)=x+4$, $n(B)=2x$, $n(A\cap B)=3$ and $n(A\cup B)=19$. Find $x$.	a. 6
   2. If $A=\{1,2,3,4\}$ and $B=\{3,4,5\}$, what is $A\cup B$?	b. 31
   3. $n(A)=12$, $n(B)=9$ and $n(A\cap B)=4$. Find $n(A\cup B)$.	c. $\{1,2,3,4,5\}$
   4. A set has 5 elements. How many proper subsets does it have?	d. 17

   Show answer

   1 → a (6) · 2 → c ($\{1,2,3,4,5\}$) · 3 → d (17) · 4 → b (31)

   • Step 1. $n(A\cup B)=n(A)+n(B)-n(A\cap B)$: $(x+4)+2x-3=19$.
     Step 2. $3x+1=19$, so $3x=18$ and $x=6$.
   • The union contains all elements of either set, each written once.
   • $n(A\cup B)=12+9-4=17$.
   • Step 1. Number of subsets: $2^5=32$.
     Step 2. A proper subset is any subset except the set itself: $32-1=31$.

4. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $\{a,b,e\}$  ·  $A=\{1,2,3\},B=\{x,y\}$  ·  $\{a,b,c,d,e\}$  ·  $\{a,b\}$
   1. $A=\{a,b,c,d\}$ and $B=\{c,d,e\}$. Find $A-B$. →
   2. $A\times B=\{(1,x),(1,y),(2,x),(2,y),(3,x),(3,y)\}$. What are $A$ and $B$? →
   3. $A=\{a,b,c,d\}$ and $B=\{c,d,e\}$. Find the symmetric difference $A\Delta B$. →
   Show answer

   1: $\{a,b\}$ · 2: $A=\{1,2,3\},B=\{x,y\}$ · 3: $\{a,b,e\}$

   • $A-B$ keeps the elements of $A$ that are not in $B$: $a$ and $b$.
   • First components form $A$, second components form $B$.
   • $A\Delta B=(A-B)\cup (B-A)=\{a,b\}\cup \{e\}=\{a,b,e\}$.

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